Internal structure of addition chains: Well-ordering

Abstract

Abstract. An addition chain for n is defined to be a sequence (a 0 , a 1 , . . . , ar) such that a 0 = 1, ar = n, and, for any 1 ≤ k ≤ r, there exist 0 ≤ i, j < k such that a k = a i + a j ; the number r is called the length of the addition chain. The shortest length among addition chains for n, called the addition chain length of n, is denoted ℓ(n). The number ℓ(n) is always at least log 2 n; in this paper we consider the difference δ ℓ (n) := ℓ(n) − log 2 n, which we call the addition chain defect. First we use this notion to show that for any n, there exists K such that for any k ≥ K, we have ℓ(2 k n) = ℓ(2 K n) + (k − K). The main result is that the set of values of δ ℓ is a well-ordered subset of [0, ∞), with order type ω ω . The results obtained here are analogous to the results for integer complexity obtained in [1] and [3]. We also prove similar well-ordering results for restricted forms of addition chain length, such as star chain length and Hansen chain length.

Integer complexity, stability, and self-similarity

“…Moreover, as also shown in [6], stabilization has its analogue as well: Theorem 1.17. For any natural number n, there exists K ≥ 0 such that, for any…”

“…then it was shown in [6] that this is also true for addition chain defects: Theorem 1.16 (Addition chain well-ordering theorem). The set…”

Integer complexity: the integer defect

“…The addition chain analogue of this is that one has ℓ(2n) ≤ ℓ(n) + 1, with equality if and only if δ ℓ (2n) = δ ℓ (n). The result [1,6] is that if we have two numbers m and n with δ ℓ (n) = δ ℓ (m), then one must have m = 2 k n for some k ∈ Z; and if we have two numbers m and n with δ(n) = δ(m), then one must have m = 3 k n for some k ∈ Z. However in the latter case we must also have m ≡ n (mod 3); this is why the sets D a are disjoint.…”

“…for as shown in [1], the well-ordering theorem for integer complexity has an analogue for addition chains: Theorem 1.15 (Addition chain well-ordering theorem). Let D ℓ denote the set {δ ℓ (n) : n ∈ N}.…”

“…But it is not only our primary theorem but also our secondary theorem here that has an analogues for addition chains, and in this case the analogy does not rely on any conjectures. While the hypothesis that the order type of D ℓ ∩ [0, k] is equal to ω k remains a conjecture, that this holds for k ≤ 2 -and in particular that it holds for k = 1 -was proven in [1]. This means that just as we can look at the first ω elements of each D a in order to determine the r'th-highest number of complexity k, we can look at the first ω elements of D ℓ to determine the r'th-highest number of addition chain length k (or at most k, which in these cases is the same thing).…”